《高数双语》课件section 1_2.pptx

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1、Section 1.2,Limits of Sequences of Numbers,Area of a Circle,2,Do you ever remember how to obtain the area of a circle?,The area of a hexagon,The area of a dodecagon,.,Area of a Circle,3,Informal Definition of Limit of a Sequence,The area of the circle is the limit of the areas of the inscribed polyg

2、ons.,A sequence can be thought as a list of numbers written in a definite order:,Note This definition is“informal”because phrases like arbitrarily close and sufficiently large are imprecise;their meaning depends on the context.,Limit of a Sequence,School of Science,BUPT,4,gets arbitrarily close to A

3、 as n becomes sufficiently large,5,Limit of a Sequence,Definition A sequence of numbers 数列 is just a special,is called the general term 通项,of the sequence.,Example,6,Limit of a Sequence,Approach to zero 趋于零,Does not approachany value,Approachesinfinity 无穷,The limits are 0,No limits,While n tends to

4、infinity,7,Limit of a Sequence,Since a sequence can be seen as a function of positive integer,we can plot a sequence on a coordinate plane.,Limit of a Sequence,8,Observe the changing of a sequence while,Limit of a Sequence,9,Observe the changing of a sequence while,Limit of a Sequence,10,Observe the

5、 changing of a sequence while,Limit of a Sequence,11,Observe the changing of a sequence while,Limit of a Sequence,12,Observe the changing of a sequence while,Limit of a Sequence,13,Observe the changing of a sequence while,Limit of a Sequence,14,Observe the changing of a sequence while,Limit of a Seq

6、uence,15,Observe the changing of a sequence while,Limit of a Sequence,16,Observe the changing of a sequence while,Limit of a Sequence,17,Observe the changing of a sequence while,Limit of a Sequence,18,Observe the changing of a sequence while,19,Limit of a Sequence,Observe the changing of a sequence

7、while,20,Concept of Limit of a Sequence,Definition(Limit of a sequence of numbers),or,If there is no such number A,we say that an diverges 发散.,holds for all,only if to every positive number there exists a positive integer N such that,to infinity,denoted by,And A is called the limit 极限 of the sequenc

8、e,Concept of Limit of a Sequence,21,Another expression of the definition of limit for a sequence is:,Note The definition can be used to judge whether A is the limit of an,but is not an efficient method used to find the limit or judge whether the limit exists.,22,Geometric explanation of the definiti

9、on,Concept of Limit of a Sequence,Let us draw the sequence an on an axis to demonstrate the definition of limit by geometric graph,23,Example,Concept of limit of a sequence,Proof For any given,we want to find a N,s.t.,Prove that,Since,we only need to find a N,s.t.,It is enough to choose,for all,.Thi

10、s is the end.,for all,So we have found a N,s.t.,Hence,The definition of limit is an important tool used to judge if a constant is the limit of a sequence.,Concept of Limit of a Sequence,24,or,25,Example,Concept of Limit of a Sequence,Proof For any given,Since,we need to find a N,s.t.,Choosing,This m

11、eans,.This is the end.,we have found a N,that,26,Concept of Limit of a Sequence,Example,Prove that,where,Proof,there is no harm in assuming that,We want to find a,such that,Notice that ln|q|0,so the requirement is,Then,for all,we have,This means,27,Concept of Limit of a Sequence,Example,Since,it is

12、enough to choose,s.t.,This is the end.,Proof For any given,Concept of Limit of a Sequence,28,Note is used to describe the fact that the distance|an-A|may be arbitrarily small.,Also,we can use,or,29,Concept of Limit of a Sequence,Example,Proof Since,we have,such that,that is,.Since,and,we have,.Notic

13、ing that,we have,Moreover,we have,.That is,So,we have,for all nN,.,.,.,Finish.,30,Concept of Limit of a Sequence,For example,the sequence,does not approach a constant.,or,31,Concept of Limit of a Sequence,or,32,Concept of Limit of a Sequence,Theorem(Uniqueness)The limit of any convergent sequence is

14、 unique.,Assume that the sequence has two different limits,and,that is,and,.,Proof,Lets suppose,.Since,by the definition of limit,for any given,there exists,such that,for all,.It means that,Similarly,since,we have,This is impossible.The proof is completed.,a,.,for some N2 and for all n N2.,B,A,(),()

15、,33,Concept of Limit of a Sequence,Theorem(Preservation of sign),34,Concept of Limit of a Sequence,Theorem(Isotone),35,Conditions for Convergence of A Sequence,Definition(boundedness of a sequence),If there exists a constant,holds for all,.Then,is said to be bounded above(or below).,such that,is sai

16、d to be bounded.,Moreover,if,Unbounded,bounded:,unbounded:,36,Conditions for Convergence of a Sequence,Theorem(boundedness)Any convergent sequence must be bounded.,is a sequence and,for given,there exists a,such that,Thus,when,we have,or,Proof,By the definition of limit,holds for all,.,Suppose that,

17、37,Conditions for convergence of a sequence,There are at most,points,outside the interval,.,that we can find a number B,s.t.,If we choose,then,This is the end.,.,is bounded.,Proof(continued),Since there are finite terms,it is obvious,Theorem(boundedness)Any convergent sequence must be bounded.,38,De

18、finition(monotonicity of a sequence),Conditions for convergence of a sequence,if we always have,A sequence which is,either monotone increasing or monotone decreasing is called monotone.,Monotone increasing,Monotone decreasing,39,Conditions for convergence of a sequence,Theorem(monotone boundedness c

19、riterion)If a sequence is monotone increasing and bounded above or monotone decreasing and bounded below,then it must be convergent.,40,Conditions for convergence of a sequence,Note It is clear that changing any finite number of terms of a sequence does not affect its convergence and limit.,For inst

20、ance,let,.,We had known that this sequence doesnot have limit or does not convergent.,.,.,If we change the first 2n numbers of this sequence to 1 and keep the others unchanged.We have,It is clear that this sequence still does not have limit or does not convergent.,41,Conditions for convergence of a

21、sequence,then the new sequence becomes convergent or has limit.,.,42,Conditions for convergence of a sequence,Note If a sequence is bounded but monotone only starting form term,then it is still convergent.,Note It should be recognized that the theorem is only a sufficient condition for convergence.A

22、 convergent sequence is not necessarily monotone.,For instance,let,43,Conditions for convergence of a sequence,Proof Since,and,by mathematical induction method,we have,Then,by monotone boundedness criterion,we can finish the proof.,for all n.,Moreover,it is clear that,for all n.,44,Conditions for co

23、nvergence of a sequence,45,Conditions for convergence of a sequence,Therefore,for any,we have,.So,If we can prove,is bounded above,then we will finish the proof.,is monotone increasing.,46,Conditions for convergence of a sequence,Proof(continued),Actually,term,can be expended as,47,Conditions for co

24、nvergence of a sequence,Remark This example shows that sequence,has limit and,.That is,this limit is denoted by,.,and,e,48,Conditions for convergence of a sequence,For example,49,Conditions for convergence of a sequence,Theorem(Cauchys convergence principle)The necessary and sufficient condition for

25、 a sequence to be convergent is that,Remark,accumulate in an arbitrarily small neighborhood of some point.,is arbitrarily small for any,and,then all terms will accumulate in an arbitrarily small,is large enough,.,all the terms will,Conversely,neighborhood of some point when,converges to this point.,

26、and the sequence,50,Conditions for Convergence of a Sequence,.,Proof We will use Cauchys theorem.,and,is equivalent to taking any,and considering,and,.Then,In fact,taking any,.,51,Conditions for convergence of a sequence,Proof(continued),we want to judge whether there exists a,s.t.,Since,then we can

27、 choose,Finish.,.,.,Hence,such that,there exists,.,52,Conditions for convergence of a sequence,Example,is divergent,where,Proof,such that,We now take,.Since,Finish.,Prove that the sequence,.,We only need to show that the condition of Cauchys theorem can not be hold.,That is,53,Conditions for converg

28、ence of a sequence,54,Conditions for convergence of a sequence,Solution,Taking n=4k,we get a subsequence,which converges to 0.,Taking n=8k+2,we get a subsequence,which converges to 1.,According to the aggregation principle,the sequence,is divergent.,55,Rules of Operations on Convergent Sequence,Theo

29、rem(Rules of rational operations),then,(2),(3),if,.,(1),Rules of Operations on Convergent Sequence,56,Solution,which will be solved later.,we have,then we have,Finish.,and we know that,That is,That is,(the negative value should be,We can suppose the limit is,if a sequence has limit,then the limit mu

30、st unique.,.,.,So,.,omitted).,57,then,(2),.,Rules of Operations on Convergent Sequence,Corollary,(1),Note,The rules(1)and(2)rational operations theorem may be extended to any,if the limit of each one exists,but they are not necessarily,true for an infinite number of sequences.,finite number of seque

31、nce,58,Rules of Operations on Convergent Sequence,Example Find,Solution Since,infinite number,59,Rules of Operations on Convergent Sequence,Solution,we obtain,Using rules of rational operations theorem and its corollary we have,Example Find,60,Example Discuss the convergence of the following sequenc

32、e and find its limit if it exists.,Solution,divergence,Rules of Operations on Convergent Sequence,61,Rules of Operations on Convergent Sequence,Solution Since,we have,Example Find,62,Example Find,and,Then,it is clear that,Since,Solution,.,Rules of Operations on Convergent sequence,63,Theorem(Sandwic

33、h Theorem 夹逼准则),such that,holds for all,and,then the sequence,is also convergent,Sandwich Theorem,If,64,Sandwich Theorem,Example Find,Solution:,and,by the Sandwich theorem we known,where,Since,?using the sandwich theorem,65,Sandwich Theorem,Solution,and,Then,by the Sandwich theorem we known,Example

34、Find,Since,.,66,Sandwich Theorem,Example Find,where a 0 is a constant.,Solution,When a=1,it is easy to obtain,When a 1,let,By the binomial formula,we have,So that,and,Then by the Sandwich theorem we known,When 0 a 1,then,so that,Hence,67,Examples,Example Find,Solution,68,Examples,Example Find,Soluti

35、on,Let,since,and,Then,by the Sandwich theorem we known,?,69,Examples,Example Find,Solution,Since,we have,70,Concept of limit of a sequencean A:uniquenessConditions for convergence of a sequenceMonotone boundedness criterionCauchys convergence principleRules of operations on convergent sequencesRules of rational operationsSandwich theorem,Review,